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Let's go through each question one by one and solve them.

### Question 1
You are asked to simplify the expression:

\[
(\sqrt{5} + 3\sqrt{2})(\sqrt{5} - 2\sqrt{2})
\]

This is a binomial product, and we can expand it using the distributive property (FOIL method):

\[
(\sqrt{5} + 3\sqrt{2})(\sqrt{5} - 2\sqrt{2}) = \sqrt{5} \cdot \sqrt{5} + \sqrt{5} \cdot (-2\sqrt{2}) + 3\sqrt{2} \cdot \sqrt{5} + 3\sqrt{2} \cdot (-2\sqrt{2})
\]

Now let's simplify each term:

1. \(\sqrt{5} \cdot \sqrt{5} = 5\)
2. \(\sqrt{5} \cdot (-2\sqrt{2}) = -2\sqrt{10}\)
3. \(3\sqrt{2} \cdot \sqrt{5} = 3\sqrt{10}\)
4. \(3\sqrt{2} \cdot (-2\sqrt{2}) = -6 \cdot 2 = -12\)

Now, add all these terms together:

\[
5 + (-2\sqrt{10}) + 3\sqrt{10} + (-12)
\]

Combine like terms:

\[
5 - 12 = -7
\]
\[
-2\sqrt{10} + 3\sqrt{10} = \sqrt{10}
\]

Thus, the expression simplifies to:

\[
\boxed{\sqrt{10} - 7}
\]

This corresponds to **Option C**.

---

### Question 2
Simplify the radical expression:

\[
\sqrt[3]{x^{15}}
\]

The property of radicals and exponents states that:

\[
\sqrt[n]{x^m} = x^{m/n}
\]

So in this case:

\[
\sqrt[3]{x^{15}} = x^{15/3} = x^5
\]

Thus, the simplified expression is:

\[
\boxed{x^5}
\]

This corresponds to **Option D**.

---

Do you want more details or have any questions?

Here are 5 related questions for practice:
1. Simplify \((\sqrt{7} + 2\sqrt{3})(\sqrt{7} - 4\sqrt{3})\).
2. Simplify \((3\sqrt{5} + \sqrt{2})(3\sqrt{5} - \sqrt{2})\).
3. Simplify \(\sqrt[4]{x^8}\).
4. Simplify \((2\sqrt{3} + 5)(2\sqrt{3} - 5)\).
5. Simplify \(\sqrt[6]{x^{12}}\).

**Tip:** When simplifying expressions involving radicals, always look for patterns like the difference of squares or the application of exponent rules to reduce the expression.

Simplify the expression (√5 + 3√2)(√5 - 2√2) and Simplify the radical ³√x^15

Learn how to simplify expressions involving binomial products with square roots and cube root radicals. This example focuses on applying the difference of squares theorem and simplifying radical exponents. Suitable for students in Grades 9-12.

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